The current best approximation ratio in this classic setting is
\(2.06-\varepsilon \)  for some fixed \(\varepsilon > 0\)  by Chawla, Makarychev, Schramm,
and Yaroslavtsev [1]}, which extended the pivot rounding
framework of [2]} with advanced functions that convert LP
values to rounding probabilities. The standard LP relaxation for Correlation Clustering has an integrality gap of 2 [3]}. The LP-based
approximation algorithms (i.e., the 4-approximation of [3]},
the 2.5-approximation algorithm of [2]}, and the
2.06-approximation algorithm of [1]}) each prove upper bounds
on the integrality gap of this LP. Furthermore, [1]} shows
that their rounding framework cannot yield an approximation ratio
better than \(2.025\) .
